the equivariant k-localization of the g-sphere spectrum · 2017-10-26 · the equivariant...

The Equivariant K-localization of the G-Sphere pectrum Kenneth Hansen

1

The Equivariant K-localization

of the G-Sphere Spectrum

Kenneth Hansen This is part 3 of my Ph.D. thesis, which I wrote at Aarhus University, Matematisk Institut, in 1991. It has appeared in Matematisk Instituts Preprint Series, no. 37, 1991. The two other parts are Oriented, Equivariant K-theory and the Sullivan Splitting The K-localizations of Some Classifying Spaces


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The purpose of this paper is to generalize Bousfield's calculation in [B79b], (4.3) of the K-localization of the sphere spectrum to the equivariant case. This is done as follows: We select an odd prime p, and then consider two different cases: I: G is a p-group, and II: G is a finite group with order prime to p. In both cases we work p-locally. In section I we describe the G Spaces 0

GQ S and ( , )qK GF . In section 2 we consider the two different cases above and define the relevant infinite G-loop space,

( , )J G p , which is to give the GK -localization of the G-sphere spectrum GS . We also define the infinite G-loop map 0( , ) : ( , )Ge G p Q S J G p→ , and the G-space Cok ( , )J G p as the homotopy fibre of ( , )e G p . In section 3 we show that in case II we have a splitting 0 0( , ) Cok ( , )GSF J G p J G p× , where GSF , 0( , )J G p and 0Cok ( , )J G p denote the G-connected covers of 0

GQ S , ( , )J G p and Cok ( , )J G p , respectively. At the moment it doesn't seem to be possible

to prove an analogous statement in case I. In section 4 we study the GK -theory of GS and of ( , )J G p . In section 5 we briefly describe the properties of equivariant Bousfield-localization with respect to a G-spectrum, and we define the GK -localization, which in a certain sense is the correct localization to use. A G-spectrum X is GK -local, if and only if for every subgroup H of G the fixed point spectrum HX is K-local. Finally, in section 6 we calculate the GK -localization of GS . I would like to thank my Ph.D. advisors Ib Madsen and Jørgen Tornehave, and also Marcel Bökstedt for invaluable help. Especially, the idea to the definition (2.1) came from Ib Madsen. 1. Preparations In this section we collect results to be used in the following. We start by defining the spaces 0

GQ S , GSF and ( , )qK GF : Definition 1.1 Let 0

GQ S be the G-loop-space lim V VSΩ , where the limit is over all G-modules in a fixed G-universe U , cf. [LMS], p.l l ff. we demand that U is a complete G-universe, i.e. that U contains countably many copies of the regular representation

[ ]GR . Let GSF be the G-connected cover of 0

GQ S , cf [El], p.277. We here let the basepoint of 0

GQ S be the identity map between G-spheres.


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Proposition 1.2 ([S70], p.62) 0

( )

( ) ( )GG G

H

Q S Q BW H+∏ and 0( )

( ) ( )GG G

H

SF Q BW H+∏ , where the product is

over all conjugacy classes ( )H of subgroups of G, and GW H is the Weyl-group ( ) /GN H H . 0 ( )GQ BW H+ is the basepoint component of ( )GQ BW H+ .

Remark 1.3 It is well known, [Sh], p.242, that the infinite G-loop-space 0

GQ S can alternatively be obtained as follows: Let GE be the category, whose objects are pairs ( , )n ρ , where n is a non-negative integer, and : nGρ → Σ is a homomorphism. The set of morphisms ( , ) ( , )m nρ → τ is empty if m n≠ , and the set of all bijections {1,..., } {1,..., }m n→ if m n= . G acts on GE as follows: G acts trivially on objects, and if : ( , ) ( , )f m nρ → τ is a morphism, and g G∈ , then gf is the morphism sending {1,..., }i m∈ to

1( )( (( ( )( ))))g f g i−τ ρ . GE has a composition given by the disjoint union , ( , ) ( , ) ( , )m n m nρ τ = + ρ τ , where : m nG +ρ τ → Σ is the composite m n m nG ρ×τ

+⎯⎯→Σ ×Σ ⎯⎯→Σ . This makes GE into a permutative G-category, and according to [Sh], thm. A', p. 255, the group completion ( )GB BΩ E of GE is an infinite G-loop-space. It follows from [Sh], p.242, that ( )GB BΩ E is G-homotopy equivalent to 0

GQ S as an infinite G-loop-space. Definition 1.4 ([FHM], (2.1)) Let q be a prime power. Let ( )G qFGL be the category, whose objects are pairs ( , )n ρ , where n is a non negative integer, and : ( )n qG Glρ → F is a homomorphism. The set of morphisms ( , ) ( , )m nρ → τ is empty if m n≠ , and is the set of all qF -

isomorphisms m nq q→F F when m n= .

G acts on ( )G qFGL like the G-action on GE , and direct sum of qF -modules makes ( )G qFGL into a permutative G-category.

The corresponding infinite G-loop-space ( ( ))G qB BΩ FGL is denoted ( , )qK GF –see e.g. [Sh], p.242, or [FHM], (0.2). Proposition 1.5 ([FHM], (3.1)) Assume ( , ) 1q G = . Let 1 2, ,..., nV V V be the irreducible qGF -modules, and let

Hom ( , )qi G i iD V V= F be the corresponding finite field. Then

1 1

( ( , )) ( ) ( [ ])n n

Gq i q

i i

K G K D K G= =∏ ∏F F


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Definition 1.6 The functor : ( )G G qP → FE GL is defined as follows. P maps rhe object ( , : )nn Gρ → Σ of GE to ( , : ( ))n qn G Glρ → F , where nΣ is embedded in ( )n qGl F as the subgroup permiting the standard basis. On morphisms we let

( : ( , ) ( , ))P f m nρ → τ the map ( ) : m nq qP f →F F mapping the i'th standard basis

vector ie to ( )f ie . P is seen to preserve the G action and the permutative structure. Definition 1.7 The infinite G-loop-map 0: ( , )G G qe Q S K G→ F is defined to be

0( ( )) : ( ) ( ( )) ( , )G G G n q qe B B P Q S B B B B K G= Ω = Ω →Ω =F FE GL We use the notation of [El] and let GO denote the category of G-orbits, i.e. the objects of GO are the transitive G-sets /G H , where H ranges over all the subgroups of G. The morphisms are all G-maps. For a G-space X, we let XΦ denote the functor

{pointed topological spaces}G →O given by ( )( / ) HX G H XΦ = . (In general, a functor {pointed topological spaces}G →O is denoted an GO -space). Φ is seen to be a functor from {G-spaces} to { GO -spaces}. We furthermore have the functor C : { GO -spaces} → {G-spaces}, which is the right adjoint to Φ in the corresponding homotopy categories, i.e. we have the natural bijection

(1.8) [ , ] [ , ]G

GX CT X T≅ Φ O [X,CTIG N (ØX,Tloc

of [El], thm. 2, where X is a G-space and T an GO -space. We describe in detail the GO -space 0( )GQ SΦ : By considering the category GE of (1.3), we see that ( )H

GE is equivalent to the category HS consisting of all finite H-sets and H-equivalences. HS splits as a category into factors /H KT , where /H K is the typical irreducible H-set, and where the product ranges over all the H-conjugacy classes ( )HK of supgroups K of H. /H KT is the full subcategory of HS consisting of the objects ( / ), 0n H K n ≥ . From (1.2) we have that

0 0

( )

( ) ( ) ( )H

H HG H H

K

Q S Q S Q BW K+∏


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where the product runs over all H-conjugacy classes of subgroups K of H. In view of the discussion above, we see that the factor ( )HQ BW K+ originates as

( ( / ) )HQ BAut H K + – recall that ( / )H HAut H K W K= . Let K H G≤ ≤ be subgroups. The projection map / /G K G H→ , which is a morphism of GO , induces the inclusion 0 0( ) ( )H K

G GQ S Q S→ . This map is described as follows: Let 1 1/S H A= , 2 2/S H A= , ..., /n nS H A= , and 1 1/T K B= , 2 2/T K B= ... ,

/m mT K B= be a complete list of the inequivalent, irreducible H-sets and Ksets, respectively. The integers ija , 1 i m≤ ≤ , 1 j n≤ ≤ , are defined by

1

( )m

HK j ij i

i

Res S a T=

≅

Furthermore, we have the group homomorphism ( ) ( )

ijij H j H j K ij i a K ir W A Aut S Aut a T W B= = → = Σ ∫

well defined up to inner automorphisms. ijr gives a functor / /j iH A K B→T T , and we

obtain an infinite loop map : ( ) ( )ij H j K iR Q BW A Q BW B→ .

The map 0 0

1 1

: ( ) ( ) ( ) ( )m n

H KG H j K i G

j i

S Q S Q BW A Q BW B Q S= =

= → =∏ ∏ is now given

by the m n× matrix:

(1.9)

11 11 12 12 1 1

21 21 22 22 2 2

1 1 2 2

n n

n nHK

m m m m mn mn

a R a R a Ra R a R a R

A

a R a R a R

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

……

…

This is seen as follows: As both S and HKA are infinite loop maps, they are determined

up to homotopy by their restrictions to 1

m

H jj

BW A=

. These restrictions coincide, and

we conclude that S and HKA are homotopic maps.

A general morphism : / /f G K G H→ in GO is of the form ( )f gH gaK= , where a G∈ is given by ( )f H aK= . It is seen that 1a Ka H− ≤ , and as there is a one to one correspondance between 1a Ka− -sets and K-sets, the induced map 0( )( )GQ S fΦ

is given by 1

1a Ka H

K a KaI A

−

− , where 1a Ka

KI−

is the m m× -matrix given as follows: Let the

irreducible K sets be 1 2{ , ,..., }mT T T , and let the irreducible 1a Ka− -sets be 1{ ,..., }mU U .

Then the ( , )i j 'th entry of 1a Ka

KI−

is 1 if iT corresponds to jU under the 1-1 correspondance above, and is zero otherwise. 2. Equivariant J-theory In this section we define equivariant J-theory. We fix an odd prime p, and we define the p-local, equivariant space ( , )J G p in two cases:


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I: when G is a p-group, and II: when G is a finite group with order G relatively prime to p.

In both cases we select a prime q, such that 2q p+ Z generates the unit group 2( / )p ×Z , and such that ( , ) 1q G = . Such an integer q exists according to Dirichlet's

theorem, cf. [Ap], (7.9). The next key definition is inspired of [MR89], thm. C: Definition 2.1 I: Let G be a p-group. Define the functor F : {G-CW-complexes}→{Abelian groups} by

( )

( ) [ ; ( )] [ ; ( ) ( , )] GW HG Hq q q G

HH G

F X X K X K K W H≠

= × ×∏F F F

where the product is over all conjugacy elasses (H) of proper subgroups H of G, where we recall that HX is canonically a GW H -space, and where GW H acts trivially on ( )qK F .

For the sake of simplicity we denote by ( )L H the GW H -space ( ) ( , )q q GK K W H×F F , when H G< , and the G-space ( )qK F , when H G= .

Thus,

( )

( ) [ ; ( )] GW HH

H

F X X L H=∏

(This notation will normally only be used in proofs.) II: Let G be a finite group of order prime to p. Define the functor F : {G-CW-complexes}→{Abelian groups} by

( )

( ) [ ; ( )] GW HHq

H

F X X K=∏ F

where the product is over all conjugacy classes (H) of subgroups H of G, and where GW H acts trivially on ( )qK F . Definition 2.2 F satisfies the requirements of the equivariant Brown representation theorem, [LMS], (1.5.11), in both case I and II. We thus get a G-space ( , )J G p representing F, i.e. (2.3) ( ) [ , ( , )]GF X X J G p≅ , for every finite, based G-CW-complex X. (See also [B65], thm. (2.8), the proof of which carries over to the equivariant case without complications.)


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Proposition 2.4 Let H be a subgroup of G. Then

( )

( , ) ( ) ( ) ( [ ])H

Hq q q H

KK H

J G p K K K W K≠

× ×∏F F F ,

if we are in case I (i.e. if G is a p-group), and

( )

( , ) ( )H

Hq

K

J G p K∏ F

in case II. In both cases the product runs over H-conjugacy classes ( )HK of subgroups K of H. Proof: We only prove the statement in case I, as the proof in case II is virtually unchanged. As [ , ] [ ( / ), ]H GX Y X G H Y+≅ ∧ for a (non-equivariant) space X, a G-space Y and a subgroup H of G, we get the following calculations:

( )

[ , ( , ) ] [ ( / ), ( , )] [ ( / ), ( )] G

G

W KH G

K

X J G p X G H J G p X G H L K+ +≅ ∧ ≅ ∧∏

( / )KG H Ø= , if K is not conjugate to a subgroup of H, while

(2.5) 1

( / ) ( ) /( ( ) )n

KG i G i

i

G H N K N H H=

= ∩

as a GW H -set, if K is conjugate to a subgroup of H: It suffices to consider the case where K is a subgroup of H. Let 1 2, ,..., nK K K be a full collection of H-conjugacy classes of subgroups of H, such that iK is G-conjugate to K. As 1( / ) { / | }KG H gH G H gKg H−= ∈ ≤ , we study the set

1* { | }G g G gKg H−= ∈ ≤ . 1 2

* * * *... nG G G G= , where 1

* { | is -conjugate to }iiG g G gKg H K−= ∈ . It is easily seen that the GW H -set * /iG H

is isomorphic to ( ) /( ( ) )G i G iN K N K H∩ , thus proving (2.5). We now see that

( )

( )

( ) ( )

[ , ( , ) ] [ ( ( ) /( ( ) ) ) : ( )]

[ (( ( ) / ) /( ( ) / ) ) : ( )]

[ ; ( ) ] [ ; ( )] [ ; ( ) ( [ ])]

G

H

G

H

H

H H

W KHG G

K

W KG H

K

W Kq q q H

K KK H

X J G p X H K N K H L K

X H K K N K K L K

X L K X K X K K W K

+

+

≠

≅ ∧ ∩ ≅

∧ ≅

≅ × ×

∏

∏

∏ ∏F F F

as it follows from (1.5). This proves the theorem. QED

Remark 2.6 If K H G≤ ≤ , and we are in case II, the inclusion

1 1

( ) ( , ) ( , ) ( )m n

H Kq q

j j

K J G p J G p K= =

→∏ ∏F F

is given by the matrix


8

11 12 1

21 22 2

1 2

n

n

m m mn

a a aa a a

a a a

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

…

…

Here the ija 's are defined as in (1.9). This follows from the proof of (2.4): Let : / /G K G Hπ → be the projection, and let : ( , ) ( , )H Ki J G p J G p→ be the inclusion. For a CW-complex X we now see that the diagram

**

[ , ( , ) ] [ ( / ); ( , )]( )

[ , ( , ) ] [ ( / ); ( , )]

H G

X

K G

X J G p X G H J G pi Id

X J G p X G K J G p

+

+

→ ∧↓ ↓ ∧ π

→ ∧

commutes. By using the fact that the splittings of ( , )HJ G p and ( , )KJ G p come from the various irreducible H- and K-sets, we get the result. Proposition 2.7 ( , )J G p is an infinite G-loop-space, both in case I and in case II. Proof: Again we only prove the statement in case I. From (1.4) it follows that ( )L H is an infinite GW H -loop space. Thus, for every

GW H -representation HU we have an HU 'th delooping ( )HUL H , such that

( ( ) )H

H

UUL HΩ and ( )L H are G-homotopy equivalent G-spaces.

Let V be a G-module. Then the V'th delooping of ( , )J G p is the representing space for the functor

( )

: [ ; ( ) ] GH

W HHV V

H

F X X L H→∏

where the fixed point representation HV is considered as a GW H -module: Let VBF denote the representing G-space for VF . Then

( )

( ) ( )

( )

[ , ] [ , ] [( ) , ( ) ]

[( ), ( ) ] [ , ( ) ]

[ , ( )] [ , ( , )]

GH

H HG G

H H

G

W HV G V G V HV V V

H

W H W HV H H VV V

H H

W HH G

H

X BF S X BF S X L H

S X L H X L H

X L H X J G p

Ω = ∧ = ∧ =

∧ = Ω =

=

∏

∏ ∏

∏

proving that ( , )VVBF J G pΩ .

QED


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Definition 2.8 Let H be a subgroup of G. We then have the functor :

G

HH G W HI →E E , where an

object of HGE is considered as an GW H -set.

Hi induces the infinite GW H -loop map 0 0( ) : ( ) ( ) ( )

G G

H HH H G G W H W Hi B Bi Q S B B B B Q S= Ω = Ω →Ω =E E

Another description of Hi is as follows: If V is a GR -module, then we have the

GW H -map

: ( , ) ( , ) ( , )H HV V H V V V V

V Hm Map S S Map S S Map S S≅ → ,

which sends the H-map : V Vf S S→ to the map :H HH V Vf S S→ . By taking the limit

over all GR -modules V, we get a GW H -map 0 0: ( )G

HG W Hm Q S Q S→ . This map m

and the map Hi from above are GW H -homotopic maps. Definition 2.9 The map 0( , ) : ( , )Ge G p Q S J G p→ is defined as follows: In case I (the p-group case), we let the H'th component of 0 0

( )

( , ) [ , ( , )] [( ) , ( )] GW HHG G

H

e G p Q S J G p Q S L H∈ =∏

be the composite

0 0( ) ( , )

( ) ( , ) ( )

w HGH

G

eiHG W H q G

j Idq q G

Q S W S K W H

K K W H L H×

⎯⎯→ ⎯⎯⎯→

⎯⎯⎯→ × =

F

F F

when H G≠ , and the composite 10 0

1( ) ( ;1) ( ) ( )Gi eHG q qQ S Q S K K L G⎯⎯→ ⎯⎯→ = =F F

when H G= . In case II we let the H'th component of 0 0

( )

( , ) [ , ( , )] [( ) , ( )] GW HHG G q

H

e G p Q S J G p Q S K∈ =∏ F

be the composite 0 0( ) ( , ) ( ,1) ( )w HGH

G

ei jHG W H q G q qQ S W S K W H K K⎯⎯→ ⎯⎯⎯→ ⎯⎯→ =F F F

Here : ( , ) ( ,1) ( )q G q qj K W H K K→ =F F F is induced by the group homomorphism

1HW G → , and 1 denotes the trivial group. Definition 2.10 The G-space Cok ( , )J G p is defined as the G-homotopy fibre of the map 0( , ) : ( , )Ge G p Q S J G p→


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Proposition 2.11 The map 0( , ) : ( , )Ge G p Q S J G p→ is an infinite G-loop map, both in case I and in case II. Proof: We have that the H'th component of ( , )e G p in both cases is an infinite GW H -loop map, as it is composed of infinite GW H -loop maps. We now proceed as in the proof of (2.7).

QED For later use we calculate the map 0( , ) : ( ) ( , )G G

Ge G p Q S J G p→ , when G is a cyclic p-group. (In the next section we will describe 0( , ) : ( , )Ge G p Q S J G pΦ Φ →Φ in the case Il, where ( , ) 1G p = ) Recall from [H91] , (4.5), that we have a map : ( ) ( [ ])qe Q BG K G+ → F defined as follows: Consider the categories GT and ( [ ])q GFGL of G-sets of the form ( /1)n G ,

0n ≥ , and projective [ ]q GF -modules, respectively. These categories have classifying spaces ( ) ( )GB B Q BG+Ω =T and ( ( [ ])) ( [ ])q qB B G K GΩ =F FGL . The functor : ( [ ])G qP G→ FT GL , sending a G-set to its permutation

representation: ( ( /1)) [ ]nqP n G G= F , gives the infinite loop map

( ) : ( ) ( ) ( ( [ ])) ( [ ])G q qe B BP Q BG B B B B G K G+= Ω = Ω →Ω =F FT GL . Proposition 2.12 Let G be a cyclic p-group; / nG p= Z . Let 0 11 ... nG G G G= ⊂ ⊂ ⊂ = be a complete list of the subgroups of G. Then the matrix of the map

0

0 0

( , ) : ( ) ( ( / ) ) ( ) ( , )n n

G G GG t s

t s

e G p Q S Q B G G L G J G p+= =

= → =∏ ∏

has the ( , )s t 'th entry ste given by

( )

0

s t

s tst

p j e e n s te p e n s t

s t

−

−

⎧ × > ≥⎪= = ≥⎨⎪ <⎩

The map j is described in (2.9). Proof: Let K H G≤ ≤ be subgroups. Consider the composite

/( )0 0 / 0 //( / / ) ) ( ) (( ) ) ( )

G HHia G H G H G H

G G G HQ B G K Q S Q S Q S+ ⎯⎯→ = ⎯⎯⎯⎯→ where a is the inclusion of the factor, cf. (1.2), and Hi is the map from (2.8). By considering this composite as the realization of the functor, which restricts a G-set of


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the form ( / )m G K to a /G H -set, we see that ( / )m G K is mapped to ( / )s tmp G H− , where tK p= and sH p= . As ( , )e G p on the fixed point sets is composition of the map above with ( )j Id e× , we get the result.

QED 3. The Sullivan splitting In this section we only consider case II, i.e. we assume that ( , ) 1G p = . p is, as usual, an odd prime. We show that we have a p-local splitting ( ) 0 ( ) 0 ( )( ) ( ( , ) ) (Cok ( , ) )G p p pSF J G p J G p× , where 0( , )J G p and 0Cok ( , )J G p denote the G-connected covers of ( , )J G p and Cok ( , )J G p , respectively, cf. [El], p.277. Proposition 3.1 Let p be a prime not dividing the order of the group G. Let X be a G-space, and let Y be a p-local infinite G-loop space. Then the map ( ) ( ): ([ , ] ) ([ , ] )

G

Gp pFix X Y X Y→ Φ Φ O

sending the G-map :f X Y→ to the GO -map ( ) : / ( : )H H HFix f G H f X Y→ is a bijection. Proof: This is essentially [LMS], (V.6.8) and (V.6.9): If ( , ) 1G p = , then

( ) ( )( )

[ , ] [ , ]G H H INVp p

H

X Y X Y≅∏ ,

where the superscript INV indicates that we are considering homotopy classes of 'invariant maps', [LMS] (V.6.5). But such an invariant homotopy class corresponds to a GO -homotopy class of GO -maps X YΦ →Φ .

QED Proposition 3.2 For a subgroup H of G the fixed point map 0 0

( ) ( ) ( ) ( ) ( )( ) ( )

( , ) : ( ) ( ) ( , )H H

H H Hp G p p q p p

K K

e G p Q S QS K J G p= → →∏ ∏ F

is given by the product of maps 0( ) ( ) ( ): ( )p p q pe QS K→ F .

Proof: Let K H G≤ ≤ be subgroups, and denote GW H by W. The composite


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0 0( ) ( ) ( ) ( ) ( )( ) ( ) ( , ) ( )WH ei ja H

H p G p W p q p q pQ BW K Q S Q S K W K+ ⎯⎯→ ⎯⎯→ ⎯⎯→ ⎯⎯→F F

is zero, if K H< , and equal to 0( ) ( ) ( ): ( )p p q pe QS K→ F , if K H= =H. This is seen by

using discrete models: The H-set ( / )n H K is mapped via W He i a to the [ ]q WF -

module [ / ]nq W KF . But the categorical analogue of j maps a [ ]q WF -module V to its

fixed point module WV . QED

We briefly review the non equivariant case, see [M77] p. 112 ff.: We have p-local infinite loop maps

(3.3) : pJ SFα → and : pe SF J→ ,

such that : p pe J Jα → is a homotopy equivalence. This gives a splitting

(3.4) Cokp pSF J J×

where Cok pJ is the homotopy fibre of : pe SF J→ . The work of Quillen in [Q], thm.7, p.585, shows that pJ can be identified with the connected cover of ( )qK F – algebraic K-theory of the finite field qF . Finally, the map : pe SF J→ is actually the connected cover of the map : ( ) ( [ ])qe Q BG K G+ → F of (1.7), with G being the trivial group. Definition 3.5 Define 0( , ) : ( , ) GG p J G p SFα → as follows: The GO -map 0( , ) : ( , ) GG p J G p SFΦα Φ →Φ is defined to be 0 0

( ) ( ) ( )

( , )( / ) : ( , ) ( )H H H

H Hq G

K K K

G p G H J G p K SF SFΦα = α = → =∏ ∏ ∏F

By using (3.1), we let 0( , ) : ( , ) GG p J G p SFα → correspond to

0( , ) [ ( , ) , ]GGG p J G p SFΦα ∈ Φ Φ O .

Theorem 3.6 ( , )e G p and ( , )G pα induces a p-local splitting 0 0( , ) Cok ( , )GSF J G p J G p×


13

Proof: This follows immediately from the fact that the composite ( , ) ( , )e G p G pα is a G- homotopy equivalence: For each subgroup H of G the fixed point map ( ( , ) ( , ))He G p G pα is simply

( )HK

e α∏ , and from (3.4) we conclude that this is a

homotopy equivalence. QED

4. GK − theory of Cok ( , )J G p In this section we show that GK -theory of Cok ( , )J G p vanishes. To this purpose we prove some results linking GK -theory of a G-Space to the K-theory of the fixed point sets. These results will be applied later on. In order to avoid finiteness assumptions, we agree to work with K-theory with coeffecients in / pZ for an odd prime p. We remark that the results (4.1) and (4.2) holds for K-theory with integral coeffecients, provided that X is a finite G-CW-complex. Lemma 4.1 Let G be a cyclic group, X a G-CW-complex with *( ) 0HK X = for every subgroup H of G. Then *( ) 0GK X = . (Here ( )GK − is reduced GK -theory – for a finite G-CW-complex X, ( )GK X is generated by differences of G-bundles _E F satisfying the following condition: For every x X∈ , the fibres xE and xF are equivalent xG -modules.) Proof: We show this using induction over the subgroups of G. Let 1H , 2H ..., nH be an ordering of these subgroups, such that if i jH H⊆ ,then i j≥ . Since G is cyclic, every

iH is normal in G, and we let iW denote the factor group / iG H . Define a filtration 1 2 ... nX X X⊆ ⊆ ⊆ of X by

1

ji

Hi

j

X X=

=∪

Since all subgroups jH are normal in G, each jHX is a G-space, and thus iX is a G-

space. We show inductively that *( ) 0G iK X = . For 1i = we have that 1H G= , and thus * * *

1( ) ( ) ( ) ( ) 0G GG GK X K X K X R G= = ⊗ = .

Assume now that *1( ) 0G iK X − = . By considering the long exact sequence

* 1 * * *1 1 1( ) ( , ) ( ) ( )G i G i i G i G iK X K X X K X K X−− − −→ → → → →

coming from the pair 1( , )i iX X − , we see that * *1( ) ( , )G i G i iK X K X X −≅ .


14

Since we have the canonical group homomorphism iG W→ , we get a G-action on the space iEW . We have that ( )H

iEW is either contactible or the empty set Ø, depending on whether H is contained in iH or not. The projection map between the pairs 1( , )i i i iEW X EW X −× × and 1( , )i iX X − is a G-homotopy equivalence, since it is an homotopy equivalence on all fixed point sets: For every subgroup jH one of the following three conditions is satisfied: a) j i< , or b) j i≥ and j iH H⊆ , or c) j i≥ and j iH H⊆/ .

If condition a) is satisfied, then 1( ) ( )j jH Hi iX X −= , and 1( / ) jH

i iX X − is

contractible. Whether jHiEW is the zero set or contractible, the set

1( / ) jHi i i iEW X EW X −× × is contractible.

If condition b) is satisfied, then ( ) jHiEW is contractible, and the claim is trivial.

If condition c) is satisfied, then ( ) jHiEW Ø= , and 1( / ) jH

i i i iEW X EW X −× × is

contractible. But 1( ) ( ) ( )j j jiH H HHi iX X X−= ∪ , and the set ( ) ji kHH HX X= is

contained in 1( ) jHiX − ; here kH is the smallest subgroup of G-containing the

subgroups iH and jH , and since k iH H⊃ , we know that k i< , and

1kH

iX X −⊆ . We thus have * * *

1 1( ) ( , ) ( , )G i G i i G i i i iK X K X X K EW X EW X− −≅ ≅ × × We note that H acts trivially on the spaces iEW and 1/i iX X − . [MR84], (1.3.12), stating that if G is an Abelian group, Γ a subgroup of G, then there is an isomorphism /( ) ( ) ( )G GK X K X RΓ Γ

Γ≅ ⊗ Γ , reduces the problem to showing that *

1( , )iW i i i iK EW X EW X −× × is zero. By using the

long exact sequence on the pair 1( , )i i i iEW X EW X −× × , we see that it suffices to show that the restriction map * *

1( ) ( )i iW i i W i iK EW X K EW X −× → × is an isomorphism.

iW acts freely on iEW , and by using [S68c] (2.1) it suffices show that the map * *

1( ) ( )i ii W i i W iK EW X K EW X −× → × is an isomorphism.

We have the homotopy commutative diagram of fibrations

1

1i i

i i

i W i i W i

i i

X X

EW X EW X

BW BW

−

−

→↓ ↓× → ×

↓ ↓=


15

Using a Mayer-Vietoris argument, we conclude that * *1( ) ( ) 0i iK X K X −= = . The

Atiyah-Hirzebruch spectral sequence and the comparision theorem for spectral sequences show that * *

1( ) ( )i ii W i i W iK EW X K EW X −× → × is an isomorphism.

QED Lemma 4.2 ([MC], cor. C.) Let G be a finite group, p an odd prime. Let X be a G-CW-complex, such that

*( ) 0CK X = for every cyclic subgroup of G. Then *( ) 0GK X = . Proposition 4.3 Let Cok ( , )J G p be the space from (2.10). Then *(Cok ( , ); / )GK J G p pZ vanishes. Proof: According to (4.l) and (4.2) it suffices to show that

0( , ) : ( ) ( , )K K KGe G p Q S J G p→ is an *( ; / )K p− Z -equivalence, when G is a cyclic

group, and K is a subgroup of G. In case I, the p-group case, we use (2.12) and the fact, that ( ) : ( ) ( ) ( , )q qj Id e Q B K K+× Γ → × ΓF F

is a *( ; / )K p− Z -equivalence, when Γ is a cyclic p-group, cf. (1191), (4.18). In the case where G is invertible in / pZ , we use the definition (3.2) and the

fact that 0: ( )qe QS K→ F is a *( ; / )K p− Z -equivalence, cf. [MM], (5.22). QED

From (2.11) we know that Cok ( , )J G p is an infinite G-loop space. We want to show that GK -theory of the corresponding G-spectrum vanishes. Lemma 4.4 Let X be an infinite G-loop space with *( ) 0HK X = for every subgroup H of G. Let V be an GR -module. Then the V'th delooping VX of X satisfies *( ) 0H

VK X = for every subgroup H. Especially, *( ) 0GK X = . Proof: This follows from (the dual) of the K-theoretical Rothenberg-Steenrod spectral sequence, cf. [Ho], p.5 (the dualization is carried through in [McC], p.242 ff):

*

2( ) ( ) *

ˆTor ( , ) ( )K X p pE K BX E∞= ⇒ =Z Z QED


16

Corollary 4.5 The GK -theory with coeffecients in / pZ of the spectra Cok ( , )J G p vanishes. Corollary 4.6 The map of G-spectra ( , ) : ( , )Ge G p S J G p→ of (2.9) is an ( ; / )GK p− Z -equivalence. We here denote the G-sphere spectrum with GS , and we denote the G-spectra corresponding to the infinite G-loop spaces ( , )J G p and Cok ( , )J G p by the same symbols, i.e. by ( , )J G p and Cok ( , )J G p 5. Equivariant Bousfield localization In this section we briefly describe the properties of equivariant Bousfield-localization. The formal development of the theory will not be done here, as the nonequivariant results of [B79a] and [B79b] immediately can be generalized to the equivariant case: We work in the category S

GHo of G-CW-spectra as described in [LMS], p.27 ff. This is the homotopy category of spectra indexed over the complete G-universe U consisting of countably many copies of the regular representation [ ]GR . Every object in S

GHo is assumed to be a G-cell spectrum, i.e. it has a decomposition into stable G-cells of the form ( ( / ) )n

GS G H +Σ ∧ , where n ranges over the integers, and H over the subgroups of G. Definition 5.1 Let A be a G-spectrum. A G-spectrum X is A-acyclic, if *( ) 0A X = . A map

:f X Y→ in SGHo is an A-equivalence, if the map * * *( ) : ( ) ( )A f A X A Y→ is an

isomorphism. The G-spectrum B is A-local if, for every A-equivalence :f X Y→ , the map *

* *:[ , ] [ , ]f Y B X B→ is an isomorphism. Finally, the map :f X Y→ is an A-localization of X, if f is an A-equivalence, and Y is A-local. Analogous to [B79b] (1.1), we have Theorem 5.2 Every G-spectrum X in S

GHo has an A-localization, denoted by AX or AL X . This A-localization is unique up to equivalence.


17

The most obvious examples of equivariant Bousfield localizations are localization with respect to Eilenberg-MacLane speetra, cf. [LMM9. Especially, if

( )( ,0)G pX H= Z , where ( )pZ denotes the constant coeffecient system ( )/ pG H Z , then XL is the localization at the prime p of [MMT]. Similarly, localization with respect to ( ,0)G qH F gives the p-adic completion of [M81]. Both these localization functors have the property that they preserve the formation of fixed point sets: Proposition 5.3 Let X be one of the spectra ( )( ,0)G pH Z or ( ,0)G qH F . Let A be any G-spectrum, and let M be a subgroup of G. Then ( ) ( )M

M MXX

L A L A Proof: This is thm. 10 of [MMT] and thm, 14 of [M81].

QED For general spectra X, the statement of (5.3) does probably not hold. We therefore introduce a variant of GK -theory, which in the equivariant case is more manageable: Definition 5.4

Let GK be the G-spectrum defined by ( )

( / )G HK G H+= ∧∨K , where the wedge is

over all conjugacy classes of subgroups of G, K is the spectrum representing ordinary K-theory, and /G H+ is the usual G-space. Proposition 5.5 1) X is GK -acyclic if and only if *( ) 0HK X = for every subgroup H of G. 2) Every GK -local G-spectrum is GK -local. Proof: 1) is obvious. In order to prove 2), let X be a GK -acyclic G-spectrum. This implies that for every subgroup H of G we have that [ ( / ), ] [ , ] 0G HK G H X K X+∧ = = . Thus HX is K-local, and *( ) 0HK X = . (3.2) now implies that ( ) 0GK X = .

QED


18

It follows immediately from the definition (5.4) that Proposition 5.6 Let X be a G-spectrum, M a subgroup of G. Then ( ( )) ( )

G

M MKL X L XK .

6. The equivariant K-localization of the G-sphere spectrum In this section we calculate the GK -localization of the equivariant sphere-spectrum GS . Again, all spectra are assumed to be p-local, and we work with the two cases: I: G is a p-group, and Il: G is a finite group with ( , ) 1p G = . Out starting point is (4.6), which we immediately generalize to p-adic coeffecients: Proposition 6.1 The map of G-spectra ( , ) : ( , )Ge G p S J G p→ of (2.9) is an ˆ( ; )G p−K Z -equivalence. Proof: By using the Bockstein sequences coming from the coeffecient sequences 10 / / / 0n np p p−→ → → →Z Z Z , we see inductively that ( , )e G p is a ( ; / )n

G p−K Z equivalence, i.e. for every subgroup H of G the map ( , ) : ( , )H H H

Ge G p S J G p→ is a ( ; / )nK p− Z -equivalence. Let X be a spectrum, and let 0S A be the Moore spectrum for the Abelian group A, see [A74], p.200. We have that 0 0ˆ lim / n

pS S p≅Z Z , and thus

* 0 0 0* * *

ˆ ˆ ˆ( ; ) [ ; ] [ ; ] [ lim / ; ]np p pK X X K S X DS K X DS p K≅ ∧ ≅ ∧ ≅ ∧Z Z Z Z ,

where DZ is the Spanier-Whitehead dual of the spectrum Z. The Milnor sequence applied on the space 0lim / nX DS p∧ Z now gives the natural exact sequence 1 * 1 * *ˆ0 lim ( ; / ) ( ; ) lim ( ; / ) 0n n

pK X p K X K X p−→ → → →Z Z Z

By applying this sequence on the map ( , ) : ( , )H H HGe G p S J G p→ , we obtain the

diagram1 * 1 * *

1 * 1 * *

ˆ0 lim ( ; / ) ( ; ) lim ( ; / ) 0

ˆ0 lim ( ( , ) ; / ) ( ( , ) ; ) lim ( ( , ) ; / ) 0

H n H H nG G p G

H n H H np

K S p K S K S p

K J G p p K J G p K J G p p

−

−

→ → → →↓ ↓ ↓

→ → → →

Z Z Z

Z Z Z


19

As the first and last vertical arrows are isomorphisms, we conclude that the middle arrow is an isomorphism, too, and the result now follows from (5.5).

QED This almost calculates the GK -localization of GS . But there are two difficulties:

( , )J G p is not GK -local G-spectrum, and ( , )e G p is not a ( ; )G − QK -equivalence. We remedy this: Recall from [FHM], (0.5), that we have a cofiber sequence

(6.2) 1( , ) 0, 2,q

q G GK G K Kψ −→ < ∞ > ⎯⎯⎯→ < ∞ >F

where ,GK n< ∞ > is the n-connected cover of the G-spectrum GK . Define ( , )q GFK

as the homotopy fibre of 1:qG GK Kψ − → .

We remark that ( , )q GFK is a GK -local G-spectrum, as GK is GK -local. Furthermore, ( , )q GFK is GK -local: For a subgroup H of G we have the cofiber sequence 1( , )

qH H Hq G GG K Kψ −→ ⎯⎯⎯→FK

It follows from [FHM], (3.1), that HGK is equivalent to

( )Irr HK∨ , where the wedge is

over all inequivalent, irreducible HC -modules. (K is the (non-equivariant) spectrum representing ordinary complex K-theory). As H

GK is K-local, ( , )Hq GFK is K-local,

and it follows that ( , )q GFK is GK -local. Proposition 6.3 The map : ( , ) ( , )q qa K G G→F FK coming from the diagram

1

1

( , ) 0, 2,

( , )

q

q

q G G

a

q G

K G K K

G K

ψ −

ψ −

→ < ∞ > ⎯⎯⎯→ < ∞ >↓ ↓ ↓

→ ⎯⎯⎯→

F

FK

is a ˆ( ; )G p−K Z -equivalence. Proof: We show that for every subgroup H of G the map : ( , ) ( , )H H H

q qa K G G→F FK is a *( ; / )K p− Z -equivalence. Let F denote the fibre of Ha . It then suffices to show that *( ; / )K F pZ vanishes. Let mF denote the m-connected cover of F. We have the sequence of maps 1 0 1 20 ...F F F F− −= → → → →


20

and lim nF F−= . As * *( ; / ) lim ( ; / )nK F p K F p−=Z Z , it suffices to show that

*( ; / ) 0nK F p− =Z . This is done inductively: For 0n < , nF− is the zero spectrum. We obtain nF− from 1nF− + via the cofiber sequence 1 ( ( ); )n n nF F H F n− + − −→ → π − where ( ( ); )nH F n−π − is the Eilenberg-MacLane spectrum with the sole non-zero homotopy group ( ( ( ); )) ( )n n nH F n F− − −π π − = π Now, * *( ( ( ); ); / ) ( ( ( ) / ; ) 0n nK H F n p K H F p n− −π − = π ⊗ − =ZZ Z , as it follows from [AH], thm. 1, and inductively we see that *( ; / ) 0nK F p− =Z .

QED Definition 6.4 Assume we are in case 1, i.e. that G is a p-group. Let ( , )G pJ be the G-spectrum representing the functor F from S

GHo to Abelian groups given by *

( )

( ) [ , ( )] GW HH

H

X X H=∏F L

Here ( )HL is the GW H -spectrum ( ) ( , )q q GW H×F FK K in case G H≠ , and ( )GL is ( )qFK .

Let : ( , ) ( , )b J G p G p→J be the map representing the natural transformation *

( ) ( )

( ) [ , ( )] [ , ( )] ( )G GaW H W HH H

H H

F X X L H X H XΠ= ⎯⎯→ =∏ ∏ L F

where a is the map ( ) ( ) ( , ) ( ) ( , ) ( )a aq q G q q GL H K K W H W H H×= × ⎯⎯→ × =F F F FK K L

when G H≠ i#11, and : ( ) ( ) ( ) ( )q qa L G K G= → =F FK L is simply a itself.

In case II, where ( , ) 1G p = , we let ( , )G pJ be the G-spectrum representing the

functor F from SGHo to Abelian groups given by

*( )

( ) [ , ( )] GW HH

H

X X H=∏F K

and where ( ) ( ,1)q qK=F FK . In this case, the map : ( , ) ( , )b J G p G p→J is representing the natural transformation *

( ) ( )

( ) [ , ( )] [ , ( )] ( )G GaW H W HH H

H H

F X X K H X H XΠ= ⎯⎯→ =∏ ∏ K F

Proposition 6.5 1) : ( , ) ( , )b J G p G p→J is a ˆ( ; )G p−K Z -equivalence, and 2) ( , )G pJ is GK -local. Proof: l) follows immediately from (6.3) by calculating the action of b at the fixed point


21

spectra – cf. (2.4). In order to show 2), let X be a GK -acyclic G-spectrum. Then we have for every subgroup H of G and every subgroup V of GW H that ( )H VX is K-acyclic. It follows from (4.1) that HX is

GW HK -acyclic, and *[ , ( ) ( , )] 0GW HHq q GX W H× =F FK K , as both

( )qFK and ( , )q GW HFK are GW HK -local. Thus *[ , ( , )] 0GX G p =J .

The proof in case II is similar. QED

We now study the rational type of GS . We recall from (1.2) that the sole non-zero rational homotopy group of GS is in dimension 0, and that 0 ( ; )GS Aπ = ⊗Q Q , where the GO -group A⊗Q is given by ( / ) ( )A G H A H⊗ = ⊗Q Q . ( )A H here denotes the Burnside ring of the finite group H. In the p-group case (case I) we have that the G-spectrum ( , )q GFK has two non-zero rational homotopy groups, as it follows from (6.2). Both 0 ( ( , ); )q Gπ F QJ and

1( ( , ); )q G−π F QJ equal ( )q

A R′⊕ ⊗F Q , where the GO -group ( )q

A R′⊕ ⊗F Q is given by

( ) ( )

( )( / )( )

q

q

A H R H H GA R G H

A G H G

⊕ ≠⎧⎪′⊕ = ⎨=⎪⎩

FF

( )

qR HF is the Grothendieck group of [ ]q HF -modules. It follows that for a subgroup

H of G the spectrum ( )

( ( , )) ( ) H

H

W KH

K

G p H= ∏J L only has non-zero rational

homotopy in dimensions –1 and 0. Similarly, in case II ( , )G pJ has non-zero rational homotopy only in dimensions –1 and 0, and both are given by A⊗Q . Definition 6.6 Let ( , )G p′J be the homotopy fibre of the map 1: ( , ) ( ( ( , )); ); 1)GI G p H G p−→ π −QJ J , which induces the identity map at 1 1( ( , ); ) ( ( , ); )G p H G p− −π =Q QJ J . (The Hurewicz map 1 1: ( ( , ); ) ( ( , ); )H G p H G p− −π →Q QJ J is an isomorphism, as it follows from [Sp], (9.6.15) – Hurewicz' theorem modulo the Serre class consisting of finite, Abelian groups.) Let ( , )e G p′ denote the lift of ( , ) : ( , )Gb e G p S G p→J to ( , )G p′J –

( , ) 0I b e G p = as 1( ) 0GS−π = and thus ( , )b e G p lifts.


22

Proposition 6.7 ( , )G p′J is a GK -local G-spectrum. ( , ) : ( , )Ge G p S G p′ ′→J is a ˆ( ; )G p−K Z -equivalence. Proof: 1( ( ( , ); ); 1)GH G p−π −QJ is a GK -local spectrum, as every rational spectrum is K-local, see [Mi], p.207. From (6.5.2) it now follows that ( , )G p′J is GK -local. Furthermore, as 1( ( ( , ); ); 1)GH G p−π −QJ vanishes p-adically, we conclude that the map ( , ) ( , )G p G p′ →J J , and hence the lifted map ( , ) : ( , )Ge G p S G p′ ′→J are

ˆ( ; )G p−K Z -equivalences. QED

Definition 6.8 Let G be a p-group. Define the GO -map 0: ( ( , )) ( )

qP G p R G′π → ⊗F QJ as

follows: For a proper subgroup H of G, ( / )P G H is the composite 0 ( ( , )) ( ) ( ) ( ) ( )

q q q

rG p A H R H R H R Hπ′π = ⊕ ⎯⎯→ ⎯⎯→ ⊗F F F QJ , where π is the projection onto the second factor, while r is the 'rationalization map'. For H G= we let ( / )P G G be the zero map. Definition 6.9 Assume G is a p-group. The GO -map ( )P G of (6.8) corresponds to a G-map : ( , ) ( ( ) ;0)

qGG p H R G′ → ⊗F QP J

Define the G-spectrum ( , )G pJ as the homotopy fibre of P . In case II, where ( , ) 1G p = , we define ( , )G pJ to be the G-spectrum

( , )G p′J of (6.6). Theorem 6.10 The ( )( ; )G p− ZK -localization of GS is ( , )G pJ . Proof: It follows from the definition of ( , )G pJ that ( , )e G p′ factors through ( , )G pJ , and that this factored map is a ( ; / )G p−K Z -equivalence. A direct inspection reveals that this factored map is a ( ; )G − QK -equivalence.

QED


23

References [A74] J.F.Adams: Stable homotopy and generalized homology. University of Chicago Press, 1974 [Ap] T.M.Apostol: Introduction to Analytic Number Theory. Springer, 1984. [AH] D.W.Anderson, L.Hodgkin: The K theory of Eilenberg MacLane complexes. Topology 7 (1968), 317-329. [B79a] A.K.Bousfield: The Boolean algebra of spectra. Comm.Math.Helv. 54 (1979), 368 377. [B79b] A.K.Bousfield: The localization of spectra with respect to homology. Topology 18 (1979), 257-281. [B65] E.H.Brown: Abstract homotopy theory. Trans. AMS 119 (1965), 79-85. [El] A.D.Elmendorf: Systems ol Fixed Point Sets. Trans.Amer.Math.Soc., 277 (1983), 275-284. [FHM] Z.Fiedorowicz,H.Hauschild,J.P.May: Algebraic K-theory. Proc.Oberwolfach 1980. Springer LNM 967 [H91] K.Hansen: The K-localizations of some classifying Spaces. Preprint, Århus, 1991. [Ho] L.H.Hodgkin and V.P.Snaith: Topics in K-theory. Springer, LNM 496. [LMM] L.G.Lewis, J.P.May and J.McClure: Ordinary ( )RO G -graded cohomology. Bull. AMS 4 (1981), 208 211. [LMS] L.G.Lewis, J.P.May and M.Steinberger: Equivariant Stable Homotopy Theory. Springer, LNM 1213. [MM] I.Madsen and R.J.Milgram: The Classifying Spaces for Surgery and Cobordism of Manifolds. Princeton University Press, 1979. [MR84] I.Madsen and M.Rothenberg: Periodic Maps of Spheres of Odd Order. Preprint, Århus, 1984. [MR89] I.Madsen and M.Rothenberg: On the Classification of G-spheres II: PL Automorphism Groups. Math.Scand. 64 (1989), 161 218. [McC] J.McCleary: User's Guide to Spectral Sequences. Publish or Perish, 1984. [MC] J.E.McClure: Restriction Maps in Equivariant K-Theory. Topology 25 (1986), 399-109. [M77] J.P.May: E∞ -Ring Spaces and E∞ -Ring Spectra. Springer, LNM 577. [M81] J.P.May: Equivariant Completion. Bull.London Math.Soc. 14 (1982) 231- 237. [MMT] J.P.May, J.McClure and G.Triantafihou: Equivariant Localization. Bull.London Math.Soc. 14 (1982) 223 230. [Mi] G.Mislin: Localization with respect to K-theory. J.Pure Appl. Alg. 10 (1977), p.201 213. [Q] D.Quillen: On the Cohomology and K-theory of the General Linear Groups over a Finite Field. Ann.Math. 96 (1972), 552-580. [S68] G.Segal: Equivariant K theory. MIES 34 (1968), 129-151. [S70] G.Segal: Equivariant Stable Homotopy Theory. Actes Congres intern. Math, 1970, Tome 2, 59-63. [S74] G.Segal: Categories and Cohomology Theories. Topology, 13 (1974), 293-312. [Sh] K.Shiuiakawa: Infinite Loop G-Spaces Associated to Monoidal Graded Categories. Publ.RIMS.Kyoto Univ 25,(1989), 239-262. [Sp] E.H.Spanier: Algebraic Topology, McGraw-Hill, 1966.

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